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Theorem onintexmid 4416
 Description: If the intersection (infimum) of an inhabited class of ordinal numbers belongs to the class, excluded middle follows. The hypothesis would be provable given excluded middle. (Contributed by Mario Carneiro and Jim Kingdon, 29-Aug-2021.)
Hypothesis
Ref Expression
onintexmid.onint ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)
Assertion
Ref Expression
onintexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem onintexmid
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prssi 3617 . . . . . 6 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ⊆ On)
2 prmg 3583 . . . . . . 7 (𝑢 ∈ On → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
32adantr 271 . . . . . 6 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ∃𝑥 𝑥 ∈ {𝑢, 𝑣})
4 zfpair2 4061 . . . . . . 7 {𝑢, 𝑣} ∈ V
5 sseq1 3062 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → (𝑦 ⊆ On ↔ {𝑢, 𝑣} ⊆ On))
6 eleq2 2158 . . . . . . . . . 10 (𝑦 = {𝑢, 𝑣} → (𝑥𝑦𝑥 ∈ {𝑢, 𝑣}))
76exbidv 1760 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → (∃𝑥 𝑥𝑦 ↔ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}))
85, 7anbi12d 458 . . . . . . . 8 (𝑦 = {𝑢, 𝑣} → ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) ↔ ({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣})))
9 inteq 3713 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
10 id 19 . . . . . . . . 9 (𝑦 = {𝑢, 𝑣} → 𝑦 = {𝑢, 𝑣})
119, 10eleq12d 2165 . . . . . . . 8 (𝑦 = {𝑢, 𝑣} → ( 𝑦𝑦 {𝑢, 𝑣} ∈ {𝑢, 𝑣}))
128, 11imbi12d 233 . . . . . . 7 (𝑦 = {𝑢, 𝑣} → (((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦) ↔ (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})))
13 onintexmid.onint . . . . . . 7 ((𝑦 ⊆ On ∧ ∃𝑥 𝑥𝑦) → 𝑦𝑦)
144, 12, 13vtocl 2687 . . . . . 6 (({𝑢, 𝑣} ⊆ On ∧ ∃𝑥 𝑥 ∈ {𝑢, 𝑣}) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})
151, 3, 14syl2anc 404 . . . . 5 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → {𝑢, 𝑣} ∈ {𝑢, 𝑣})
16 elpri 3489 . . . . 5 ( {𝑢, 𝑣} ∈ {𝑢, 𝑣} → ( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣))
1715, 16syl 14 . . . 4 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → ( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣))
18 incom 3207 . . . . . . 7 (𝑣𝑢) = (𝑢𝑣)
1918eqeq1i 2102 . . . . . 6 ((𝑣𝑢) = 𝑢 ↔ (𝑢𝑣) = 𝑢)
20 dfss1 3219 . . . . . 6 (𝑢𝑣 ↔ (𝑣𝑢) = 𝑢)
21 vex 2636 . . . . . . . 8 𝑢 ∈ V
22 vex 2636 . . . . . . . 8 𝑣 ∈ V
2321, 22intpr 3742 . . . . . . 7 {𝑢, 𝑣} = (𝑢𝑣)
2423eqeq1i 2102 . . . . . 6 ( {𝑢, 𝑣} = 𝑢 ↔ (𝑢𝑣) = 𝑢)
2519, 20, 243bitr4ri 212 . . . . 5 ( {𝑢, 𝑣} = 𝑢𝑢𝑣)
2623eqeq1i 2102 . . . . . 6 ( {𝑢, 𝑣} = 𝑣 ↔ (𝑢𝑣) = 𝑣)
27 dfss1 3219 . . . . . 6 (𝑣𝑢 ↔ (𝑢𝑣) = 𝑣)
2826, 27bitr4i 186 . . . . 5 ( {𝑢, 𝑣} = 𝑣𝑣𝑢)
2925, 28orbi12i 719 . . . 4 (( {𝑢, 𝑣} = 𝑢 {𝑢, 𝑣} = 𝑣) ↔ (𝑢𝑣𝑣𝑢))
3017, 29sylib 121 . . 3 ((𝑢 ∈ On ∧ 𝑣 ∈ On) → (𝑢𝑣𝑣𝑢))
3130rgen2a 2440 . 2 𝑢 ∈ On ∀𝑣 ∈ On (𝑢𝑣𝑣𝑢)
3231ordtri2or2exmid 4415 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 667   = wceq 1296  ∃wex 1433   ∈ wcel 1445   ∩ cin 3012   ⊆ wss 3013  {cpr 3467  ∩ cint 3710  Oncon0 4214 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-tr 3959  df-iord 4217  df-on 4219  df-suc 4222 This theorem is referenced by: (None)
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